Why is gauge invariance important in quantum mechanics?

[Heidi]

I bought a paper written by Kuo-Ho Yang (gauge transformations and QM) in which i read this:

since Lamb [1] asserted that in order to describe the interaction of a bound system with external time-varying fields to the lowest order and in the long wavelength approximation one should use an interaction perturbation of the form eE.r and not --A-p, atomic physicists have been constrained to work in one particular gauge.

the Lamb paper is not free. does it mean that we have to use only the electric field and not the vector potential A (giving the magnetic field)? what is this gauge?

the lamb paoer (1952) is fine structure in hydrogen atoms.

 

[DrDu]

This transformation is known as the Power Zienau Woolley transformation.
https://iopscience.iop.org/article/10.1088/0022-3700/7/4/023
Interestingly there are still open points:
https://www.nature.com/articles/s41598-017-11076-5
https://www.nature.com/articles/s41598-021-94405-z

 

[vanhees71]

I've only glanced over it, but I think the 3rd of the papers is correct, at least in the analysis of the choice of the Coulomb gauge fixed Hamiltonian, which is a valid description, because there the gauge is completely fixed and only the physical degrees of freedom are kept, i.e., for the fields only the transverse $\vec{A}_C$ (two field degrees of freedom).

The question now is, how to do perturbation theory in a gauge-independent way, which is the subject of Lamb's famous paper

https://journals.aps.org/pr/abstract/10.1103/PhysRev.85.259

For a very thorough discussion of these issues see

K.-H. Yang, Gauge transformations and quantum mechanics
I. Gauge invariant interpretation of quantum mechanics,
Annals of Physics 101, 62 (1976),
https://doi.org/10.1016/0003-4916(76)90275-X

K.-H. Yang, Gauge transformations and quantum mechanics
II. Physical interpretation of classical gauge transformations,
Annals of Physics 101, 97 (1976),
https://doi.org/10.1016/0003-4916(76)90276-1

I don't think that there are really "open questions". For sure Lamb was right with his observation that for the interaction Hamiltonian one "should use $\vec{E} \cdot \vec{j}$ rather than $j_{\mu} A^{\mu}$, as Yang says in his paper I.

 

[Heidi]

My question was about the primary role that is given to the electric field in the Kuo-Ho first paper about gauge transformations.
he writes:
If we assume that the occupation probabilities which result from the conventional procedure are the physically significant ones then we know that Hb^g must reduce to the form of Hb^0 in the gauge which has A(r, t) = 0 (Lamb's paper)
If A is nul we have no magnetic field.
if we start with a particle in a static electromagnetic field how can a gauge choice make the magnetic field disappear.
i know that something is wrong in my question but i am puzzled.

 

[vanhees71]

The point is you must provide physical meaningful occupation probabilities, i.e., such describing probabilities for the outcome of measurements, i.e., observables. These must be gauge independent quantities. See the really very well written 1st paper by Yang quoted in #3, where this is very nicely derived and explained in detail.

It's of course clear that a magnetic field cannot be gauged away, because the components of the electromagnetic field are gauge invariant, i.e., if there is a magnetic field present when described with potentials in one gauge, it won't change when transforming to another gauge.

Gauge invariance reflects a redundance in the description of a physical quantity. Take the free electromagnetic field, which has 2 independent field-degrees of freedom, described by the transverse electric field (the magnetic field then is also determined) or equivalently with $\Phi=0$ and the transverse vector potential ("radiation gauge").

 

[DrDu]

The PZW-Hamiltonian is expressed only in terms of fields, so I would expect any perturbation theory based on it to be gauge invariant.

 

[Heidi]

I agree with that.
In the PZW Hamiltonian there E terms and B terms.
Kuo-Ho Yang begins with a charged particle in a E field. he says after that if there is also a magnetic field some terms are to be added but it seems that all along he consider only the electric field.

He writes
If we assume that the occupation probabilities which result from the conventional procedure are the physically significant ones then we know that Hg must reduce to the form of H° in the gauge which has A(r, t) = 0 [Lamb].

this the point that puzzles me. Does he say that only electric field can verify the equations with causing no problem?

 

[DrDu]

I just want to remark, that the current and charge density is invariant under a transformation P <- P+rot Q and $M \leftarrow M -\partial Q/\partial t + \mathrm{grad}(Q_0)$, where Q is an arbitrary vector and Q_0 a scalar. Evidently, this can be used to either make P rotation free and M gradient free, or to anihilate M completely, as is often conventional in optics.

 

[vanhees71]

I agree with that.
In the PZW Hamiltonian there E terms and B terms.
Kuo-Ho Yang begins with a charged particle in a E field. he says after that if there is also a magnetic field some terms are to be added but it seems that all along he consider only the electric field.

He writes
If we assume that the occupation probabilities which result from the conventional procedure are the physically significant ones then we know that Hg must reduce to the form of H° in the gauge which has A(r, t) = 0 [Lamb].

this the point that puzzles me. Does he say that only electric field can verify the equations with causing no problem?

The point is what you take as $\hat{H}_0$ in perturbation theory to define the energy eigenbasis. To be able to interpret these as physical states $\hat{H}_0$ should be chosen gauge invariant such that the corresponding occupation probablities refer to physically observable states. Yang et al also quote an important paper by Lamb, where he shows that the electric dipole approximation is the right choice for the perturbative part $\hat{H}_1$ if you want to use $\hat{H}_0=\hat{\vec{p}}^2/(2m)$. Lamb also has verified this in an experiment. If needed, I can try to find the citations.

Another very pedagogical paper, summarizing Yang's gauge-invariant treatment is

D. H. Kobe and A. L. Smirl, Gauge invariant formulation of the interaction of electromagnetic radiation and matter, Am. Jour. Phys. 46, 624 (1978),
https://doi.org/10.1119/1.11264

 

[Heidi]

Merry Christmas to all Physics forumers.

Let me explain why i am not entirely satisfied with this article.
I opened a previous thread about a simple situation: a charged particle in magnetic fiels (no time dependence, no electric field)
I saw that there two main gauge choices: the Landau gauge and the symmetric gauge
one is with the vector potential (0,xB,0) and the othe (-yB/2,xB/2)
the first is weird with its dagneracy usign plane waves and the second seems more physical (two oscillators)

when i read that in the paper:
Taking this point of view we wish to formulate a mathematical description of this system using only operators which have gauge invariant expectation values and whose physical interpretation is also gauge invariant. We formalize our rules of interpretation by postulating a physical correspondence principle. "A quantum mechanical operator can represent a physical quantity with with classical analogue only if there exists a Newtonian quantity whose equation of motion is formally identical to the equation of motion of the operator, and if such a correspondence exists, the operator is interpreted as the quantum equivalent of the Newtonian quantity

i thought that i could find in the paper why one the two gauge seems more physical.
But no. this simple case is not dimply treated, just the precessing magnetic field and in that case he writes: Rather than provide a complete solution, which has been presented elsewhere...

is it in the second paper? and will i find in it the answer of my question?

 

[vanhees71]

Merry Christmas to all Physics forumers.

Let me explain why i am not entirely satisfied with this article.
I opened a previous thread about a simple situation: a charged particle in magnetic fiels (no time dependence, no electric field)
I saw that there two main gauge choices: the Landau gauge and the symmetric gauge
one is with the vector potential (0,xB,0) and the othe (-yB/2,xB/2)
the first is weird with its dagneracy usign plane waves and the second seems more physical (two oscillators)

The point is not in which gauge you work but to find a physically easily interpretable set of energy eigenvectors. It's crucial to understand that the observable quantities and probabilities for the outcome of measurements should refer to gauge-invariant quantities and to prove that the corresponding probabilities, expectation values, etc. are indeed gauge independent.

For the particle in a static, homogeneous magnetic field you can choose the Hamiltonian, which can be shown to be gauge invariant (see also the papers by Yang and Kobe quoted above). Now the usual treatment is to choose the compatible momentum components to get a uniquely defined complete orthonormalized system (CONS) of energy eigenstates, but these do not have a gauge-invariant meaning, and it may be complicated to interpret their physical meaning, although of course you can use them to calculate any meaningful probability or expectation value related to gauge-invariant quantities.

Alternatively you can choose a CONS of simultaneous eigenvectors of only gauge-invariant observables, leading to a easily interpretable set of energy eigenvectors. Instead of using the gauge-dependent canonical momenta $\hat{\vec{p}}$ you can consider the "kinetic momentum"
$$\hat{\vec{\pi}}=\hat{\vec{p}}-q \vec{A}(\hat{\vec{x}})=m \hat{\vec{v}}=\frac{m}{\mathrm{i} \hbar} [\hat{\vec{x}},\hat{H}],$$
where $\vec{A}$ may be chosen in any gauge you like. The commutation relations of the kinetic momenta are also gauge invariant of course,
$$[\hat{\pi}_j,\hat{\pi}_k]=\mathrm{i} q \epsilon_{jkl} B_l.$$
For the following I make $\vec{B}=B \vec{e}_3$ ($B=\text{const}$, because we consider a homogeneous magnetic field). Since
$$\hat{H}=\frac{1}{2m} \hat{\vec{\pi}}^2.$$
We can choose as a complete compatible set of (gauge-invariant!) observables to get a unique gauge-invariant CONS of energy eigenvalues: $\hat{H}$, $\hat{H}_{\text{perp}}$, and $\hat{\pi}_3$, where
$$\hat{H}_{\text{perp]}=\frac{1}{2m} (\hat{\pi}_1^2+\hat{\pi}_2^2).$$
The solution can be obtained completely algebraically with appropriate ladder operators as for the harmonic oscillator.

For a complete treatment, see my QM manuscript. I hope that's understandable although it's written in German (the formula density is high and many intermediate step of the calculation are worked out explicitly):

https://itp.uni-frankfurt.de/~hees/publ/theo3-l3.pdf

Sect. 3.11.5 (p. 88ff).

when i read that in the paper:
Taking this point of view we wish to formulate a mathematical description of this system using only operators which have gauge invariant expectation values and whose physical interpretation is also gauge invariant. We formalize our rules of interpretation by postulating a physical correspondence principle. "A quantum mechanical operator can represent a physical quantity with with classical analogue only if there exists a Newtonian quantity whose equation of motion is formally identical to the equation of motion of the operator, and if such a correspondence exists, the operator is interpreted as the quantum equivalent of the Newtonian quantity

i thought that i could find in the paper why one the two gauge seems more physical.
But no. this simple case is not dimply treated, just the precessing magnetic field and in that case he writes: Rather than provide a complete solution, which has been presented elsewhere...

Neither gauge is more physical than any other. All are equivalent, but one must keep in mind that nothing gauge-dependent can be taken as an observable, although you can of course use a gauge-dependent basis to calculate gauge-independent physical quantities.

is it in the second paper? and will i find in it the answer of my question?

I hope, the section in my manuscript answers your question. Maybe I'll translate it as another FAQ article :-).

A very good source is also the textbook by Cohen-Tannoudji and Laloe. He has a very concise section and appendix concerning these questions about gauge invariance and all that. It's of course a bit obscured in its application to the non-relativistic case, but it can be worked out carefully of course, since gauge invariance is consistent with the non-relativistic approximation for the matter part.

Also note that all these problems do not occur when using time-dependent perturbation theory, because the time-dependent Schrödinger equation is of course gauge covariant, and an expansion in powers of $\hbar$ for gauge-invariant observable quantities leads to gauge-invariant results about physical observables (as in standard QED or NRQED).